A mathematical model of catalyst combination design and temperature control in the preparation of C4 olefins through ethanol coupling

The preparation of C4 olefins through ethanol catalytic coupling is a crucial area of study. According to the experimental data obtained by a chemical laboratory for different catalysts at different temperatures, three mathematical models were developed to provide insights into the relationships among ethanol conversion rate, C4 olefins selectivity, yield, catalyst combination, and temperature. The first model is a nonlinear fitting function that analyses the relationships among ethanol conversion rate, C4 olefins selectivity, and temperature under varying catalyst combinations. Two-factor analysis of variance was employed to determine the influence of catalyst combinations and temperatures on ethanol conversion rate and C4 olefins selectivity. The second model is a multivariate nonlinear regression model that describes the relationships among the yield of C4 olefins, catalyst combination, and temperature. Finally, an optimization model was derived based on the experimental conditions; it provides a solution for the selection of the optimal catalyst combinations and temperatures to achieve the maximum yield of C4 olefins. This work has significant implications for the field of chemistry and the production of C4 olefins.


Introduction
Ethanol is a clean, easily obtained raw material that chemical industry methods and biological fermentation techniques can produce. Biological fermentation is the popular primary technology and uses corn, sugarcane, and other crops as raw materials to produce ethanol. 1 As the technology matures and the scale of use expands, the industrial application of ethanol as a raw material is also increasing. For example, ethanol is a renewable fuel that can be used in engines. 2 It can also be used as a coolant in various metal-organic frames and similar applications, 3 which has broad prospects for producing highvalue-added products, such as C 4 olens, an essential primary chemical raw material. C 4 olens can be obtained through uid catalytic cracking (FCC) or from byproducts in ethylene cracking reactions. Isobutene can be converted into methyl tertbutyl ether (MTBE) by methanol etherication, 4,5 and is one gasoline additive. The reaction mechanism of preparation of C 4 olens by ethanol coupling is the Prince mechanism, 6 or aldol condensation mechanism. 7 The preparation of C 4 olens through ethanol coupling is very complicated, and the mechanism of the reaction must be further studied. In the preparation of C 4 olens through ethanol coupling, it is crucial to control the temperature and catalyst design. 8 In an experiment in China, Lv 9 designed a Co/SiO 2 -HAP catalyst with both acid and base activities on the surface that is aimed at the preparation of C 4 olens using ethanol. She studied the optimum conditions for the catalyst charging ratio and reaction temperature in a chemical experiment; her conclusions are consistent with those of this article which employs a mathematical modelling method. Ge 10 studied the selective superposition process of mixed C 4 olens using experimental methods and investigated the inuence of reaction conditions on the selective superposition of mixed C 4 olens, such as temperature, air speed, and pressure. Through such experiments, it has been concluded that the selectivity of C 4 olens will be signicantly reduced if the temperature drops, which supports the ndings obtained from the analysis of experimental data in this paper.
However, using experimental data, the mathematical modelling method can be employed to study the quantitative relationship and optimal design in the preparation of C 4 olens through ethanol coupling, which is an interdisciplinary method. Mathematical modelling is widely used in various elds. For example, it has been applied to identify an optimization strategy to improve the performance of microbial fuel cells 11 and to assess the risk of airborne transmission of COVID-19. 12 Moreover, it has been used for drug discovery and development. 13 In examining the preparation of C 4 olens by ethanol coupling, Li et al. 14 established the Analytic Hierarchy Process/ Entropy Weight Method-Technique for Order Preference by Similarity to Ideal Solution (AHP/EWM-TOPSIS) and built a production-quality C 4 olens assessment system. With the support of the evaluation system, the improved mixed congruence method was used to simulate the production conditions of the preparation of C 4 olens through ethanol coupling and to construct the reverse neural network (BPNN). Then, the optimal scoring production scheme at different temperatures was determined using the mathematical model. Wang et al. 15 employed a logistic regression model to analyse the relationship between ethanol temperature and conversion rate with C 4 olens selectivity in C 4 olens preparation through ethanol coupling. The relationship between different catalysts and temperature with the maximum yield of C 4 olens was also examined by constructing a neural network. Zhang et al. 16 conducted a two-dimensional visualisation analysis using experimental data on ethanol-coupled C 4 olens and used clustering analysis for different catalyst combinations. Finally, a BPNN was used to calculate the reaction conditions for the maximum yield of ethanol-coupled C 4 olens. However, these studies have all been conducted from a single point of view, giving us an incomplete and unsystematic understanding of the preparation of C 4 olens by ethanol coupling.
Therefore, based on the experimental data collected from the reactions of preparing C 4 olens through ethanol coupling, this paper systematically analysed and solved the four-part problem using mathematical modelling. In the rst part, based on the characteristics of the experimental data and on the premise of the unknown reaction mechanism, the relationships among the key components, such as ethanol, C 4 olens, and temperature, were analysed, and different tting functions were compared. In the second part, a specic constant catalyst combination and reaction temperature were selected to study the data characteristics of specic indexes of the reaction components under different experimental time, which has further explained how the reaction conditions change over time. In the third part, the inuence of varying catalyst combinations and temperatures on the critical indexes of ethanol conversion rate and C 4 olens selectivity were analysed using experimental data. In the fourth part, the yield of C 4 olens in the reaction was calculated according to the experimental data, and a multivariate nonlinear model of C 4 olens yield with catalyst and temperature was established. A reasonable optimisation model was established to nd the optimum catalyst combination and corresponding temperature under different charging methods. The general reaction process of preparing C 4 olens through ethanol coupling is as follows: Ethanol þ catalyst ! combination Temperature C 4 olefinsðmain productÞ þ by À products A chemical laboratory has conducted several experiments on the preparation of C 4 olens through ethanol coupling. The corresponding experimental data were obtained by changing the experimental conditions of catalyst combination (Co loading, Co/SiO 2 , HAP loading ratio, ethanol concentration) and temperature. In Experimental Data 1, there are 21 groups of catalyst combinations (14 groups of class A, 7 groups of class B). Each group contains ve temperatures and the corresponding ethanol conversion rates, ethylene selectivity, C 4 olens selectivity, acetaldehyde selectivity, carbon number 4-12 fatty alcohol selectivity, methyl benzaldehyde and methyl benzyl alcohol selectivity, and the experimental data for the selectivity of other products. Experimental Data 2 comprises data of unknown catalyst combinations at 350°C at six time points and contains the ethanol conversion rate, C 4 olens selectivity, and so on. It is of great practical signicance to study the inuence of changing temporal conditions on C 4 olens selectivity and C 4 olens yield. It is also important to use existing experimental data and results to analyse and explore the reactions of C 4 olens preparation through ethanol coupling.

Data sources
The original experimental data used in this paper are from Question B of the 2021 Higher Education Community Cup National Mathematical Contest in Modelling for College Students; 17 charging method I was used in catalyst experiments A1-A14, and charging method II was used in catalyst experiments B1-B7. Some experimental data are shown in Tables 1 and 2, and the parameters used in this paper are presented in Table 3.
The relationships among ethanol conversion rate, selectivity of C 4 olefins, and temperature under each catalyst combination The relationships among temperature change and selectivity of ethanol conversion rate and C 4 olens are studied in different catalyst combinations. The experimental data in Tables 1 and 2 have been preliminarily analysed using scatterplots. The ndings indicate that the temperature changes in different catalyst combinations have some relationships with ethanol conversion rate and C 4 olens selectivity. The curve tting toolbox (cool) Catalyst combination number using charging method I in Table 1 Bi Catalyst combination number using charging method II in Table 1 Ethanol conversion rate, corresponding to a specic catalyst combination and temperature (%) P(T) C 4 olens selectivity, corresponding to a specic catalyst combination and temperature (%) y I Yield of C 4 olens in charging method I y II Yield of C 4 olens in charging method II x1 Co load x2 Co/SiO 2 x3 HAP x4 Amount of ethanol added per minute  in MATLAB was used for preliminary data tting. 18 Through comparing the coefficients of determination, R 2 , and the residuals among various tting functions, a better tting function was obtained. Then their correlation was analyzed. 19 Next, the data in Table 2 were classied. Since the time data are not uniformly distributed but are complete, spline interpolation was used to supplement the complete time data. The selectivity data were analysed using scatterplots and were processed according to the data trends.
Model 1: nonlinear curve tting of ethanol conversion rate, C 4 olens, and temperature The original experimental data in Table 1 suggest that the temperature increases from 250°C in each group of catalysts.
There are specic changes in the ethanol conversion rate Y and C 4 olens selectivity P, which were the core elements of the experiment. MATLAB soware was used to draw each catalyst combination scatter plot of temperature and ethanol conversion rate. For example, the relationship between temperature T and ethanol conversion rate Y in catalyst group A1 is shown in Fig. 1. The preliminary analysis of the gure indicates a specic relationship between the temperature T of the A1 catalyst and ethanol conversion rate Y; the curve tting toolbox in MATLAB    Table 5.  Table 2.  was used for tting. In the chemical reaction with an unknown mechanism, the most suitable curve model was selected according to the data distribution in the scatter plot. 20 The known values increased in the change of temperature-toethanol conversion rate, which accorded with the exponential model. However, the ethanol conversion rate is unlikely to grow explosively, as in an exponential model, and it is unlikely to exceed or equal 100%, so the exponential model was not adopted. At the beginning the trend of ethanol conversion rate increases with the temperature, and then at a certain point of time, it decreases, and it does not change periodically hence. 21 Therefore, the relation equation should be obtained by tting the Gaussian distribution model; 22 the same is true for the selectivity of C 4 olens (Fig. 2).   According to the curve tting, the relationship between temperature T and ethanol conversion rate Y satised the equation: Y ðTÞ ¼ 315:6 e À T À 569:3 149:2 2 (2) The residual value is 13.6410, and the coefficient of determination, R 2 , is 0.9817, indicating an excellent t.
By comparing the R 2 and residual values, the closer R 2 is to 1, the better, and the smaller the residual value is, the better. Furthermore, considering the simplicity of the equation, the tting functions of ethanol conversion rate, C 4 olens selectivity, and temperature under the other catalyst groups (groups A02-A14 and B01-B07) could be obtained, as shown in Table 4.
According to the tting functions in Table 4, the corresponding values of ethanol conversion rate and C 4 olens conversion at a given temperature under each catalyst combination can be calculated.
Using the data in Table 2, the experimental results under different experiment times were analysed with a specic constant catalyst and 350°C constant temperature. However, the experiment time in Table 2 is not equally spaced, and the data do not accord with the basic principles of the experiment, so the analysis could not be completed. Therefore, primary treatment should be completed for the data. Using spline interpolation, 23 starting from 20 minutes, ethanol conversion rates and selectivity indexes were calculated at an isometric time point every 20 minutes. The results are shown in Table 5.
The results presented in Table 5 suggest that the ethanol conversion rate decreases monotonically with time, and acetaldehyde selectivity increases with time. The other data uctuate around their means. The grey prediction model GM (1,1) 24,25 could describe the relationship between time and ethanol conversion rate. It was used to predict the ethanol conversion rate (%); the results are shown in Fig. 3. Fig. 3 illustrates that the ethanol conversion rate decreased with the increase in reaction time, but the rate of decline also decreased over time. It stabilized at about 29% when the reaction time was 260 minutes.
Analysis of ethylene selectivity. The ethylene selectivity values in Table 5 uctuate around the mean and are believed to follow a normal distribution, so the qq diagram (Fig. 4) was    used for verication. 26 The distribution of the data points in Fig. 4 is roughly linear, so it can be assumed that the sample data on ethylene selectivity follow a normal distribution, with a mean of 4.51 and a standard deviation of 0. 19.
Analysis of C 4 olens selectivity. C 4 olens selectivity was believed to follow a normal distribution, and the qq plot was used for verication (Fig. 5). The plot appeared linear, so the assumption of normality for the sample data for C 4 olens selectivity was supported; the mean is 38.95 and the standard deviation is 1.17.
Analysis of acetaldehyde selectivity. The grey prediction model GM (1,1) was used to predict acetaldehyde selectivity (%), and the results are shown in Fig. 6. Here, acetaldehyde selectivity increases with the increase in reaction time, but the rate of change decreases and tends to a stable value of about 9%.
Based on the above analyses, it is clear that ethylene selectivity and C 4 olens selectivity are weakly correlated with reaction time.
Effects of catalyst combinations and temperature on ethanol conversion rate and selectivity of C 4 olens. Table 1 shows that each ethanol conversion rate and C 4 olens selectivity are related to different catalyst combinations and temperatures, but the temperature range varies by catalyst. Therefore, the temperature range must be unied before analysis and processing. According to the relationship between temperature and ethanol conversion rate and C 4 olens selectivity in different catalyst combinations (obtained using the tting function in Table 4), the data corresponding to the range 250-400°C in each group of catalysts were used. For example, using the tting function in Table 4, the tting function between ethanol conversion rate and temperature under catalyst combination A1 is as follows: The data for ethanol conversion rate and catalyst combinations at a uniform temperature were obtained. Effects of different catalyst combinations and temperatures on ethanol conversion rate. The above catalyst combinationtemperature-ethanol conversion rate data were imported into MATLAB, and a box diagram was created (Fig. 7).
The mean, maximum, and minimum values of the ethanol conversion rate differ for the 21 catalyst combinations and corresponding temperatures. Moreover, the ethanol conversion rate in charging method I is higher than that in charging method II, which implies that the ethanol conversion rate may be affected by the catalyst combination, temperature, and charging method. To further verify these observations, a twofactor analysis of variance was conducted. 27 The null hypothesis of no relationship was rejected, as catalyst combination and temperature have a signicant effect on ethanol conversion rate (p < 0.001).
Additional analyses were conducted to explore the inuence of each temperature group on the ethanol conversion rate using the data in Table 6. A box plot of the ethanol conversion rate for six temperature groups was drawn, as shown in Fig. 8.
As indicated in Fig. 8, the ethanol conversion rate is the highest when temperature is high (t = 400°C) and catalyst combination A2 is used. Using two-dimensional interpolation, 28 the curves for ethanol conversion rate, catalyst combination, and temperature were obtained ( Fig. 9 and 10). Fig. 9 depicts the surface plot of the ethanol conversion rate with catalyst combination and temperature, while Fig. 10 shows the contour plot of the ethanol conversion rate with catalyst combination and temperature. From these illustrations, it is clear that the ethanol conversion rate was highest when the temperature was 400°C and the catalyst combination was A1, A3, or A6.
Effects of catalyst combinations and temperatures on the selectivity of C 4 olens. The effects of different catalyst combinations and temperatures on the selectivity of C 4 olens were analysed using the same approach. The catalyst combinationtemperature-C 4 olens selectivity box plot is shown in Fig. 11.
The results of a two-factor analysis of variance indicate that the null hypothesis that catalyst combination and temperature have no signicant effects on C 4 olens selectivity should be rejected (p < 0.001 for both). 29 As shown in Fig. 12, when the temperature increased, the C 4 olens selectivity also increased. When the maximum temperature was 400°C, the ethanol conversion rate was highest.
Two-dimensional interpolation was used to create the surface plot (Fig. 13) and contour plot (Fig. 14) for C 4 olens selectivity with catalyst combination and temperature. The results indicate that the selectivity of C 4 olens is higher when the temperature is 400°C and the catalyst combination is A2 or A3.
Analysis of the relationship between C 4 olens yield with catalyst combination and temperature. The yield of C 4 olens is the key index in the preparation of C 4 olens by ethanol coupling, and the value is equal to the ethanol conversion rate multiplied by the selectivity of C 4 olens. The previous analysis showed that catalyst combination and temperature signicantly affect the ethanol conversion rate and C 4 olens selectivity. Therefore, the catalyst combination and temperature also have a corresponding effect on the C 4 olens yield. The quantitative relationship between them was further investigated, and the regression model for C 4 olens yield, catalyst combination, and temperature was established. Using eqn (4) and available data, the yield of C 4 olens was calculated, as shown in Table 7.
The data were normalized, and a multiple linear regression model was established, with the yield of C 4 olens as the response variable, and with temperature and four catalysts (Co load, Co/SiO 2 , HAP, ethanol addition per minute) as the predictor variables. The coefficient of determination, R 2 , of the multiple linear regression 30 is only 0.69, which is small, and the optimization results are poor.
Multivariate nonlinear regression using interaction terms. According to the results of the multiple linear regression, 31 it was necessary to analyse the possible nonlinear relationship including an interaction effect between the reaction conditions. 32 Since the units of temperature, Co loading, Co/SiO 2 , HAP, and ethanol added per minute differ (Table 8), the data for these variables were divided by the corresponding data in the rst row in order to remove the units. From previous analyses, multiple  interaction effects are known to exist under charging method I, and multiple nonlinear regression was used. 33 The model can be written as follows: The R 2 value is 0.91, indicating that the interaction effects and data nonlinearity in the reaction have a strong t; however, the model is very complex, which is not conducive to interpreting the results. Therefore, stepwise regression was carried out to further highlight the model's key factors (Fig. 15). 34 The model for the stepwise regression is: (6) The results indicate that R 2 is 0.85, further highlighting the key inuencing factors and improving the applicability of the model. Moreover, it is concise.
In charging method II, rst, based on the results of the multiple linear regression and considering the existence of the interaction effects, group B1 was taken as the benchmark for comparison aer removing the units. Complete quadratic polynomial tting was used to obtain the following model: The results indicate that the R 2 value is 0.96. Furthermore, the model using stepwise regression was as follows: y II = 115.486 − 181.657 × T − 11.557 × x2 + 69.48 × T 2 + 10.4993 × T × x2 (8) The R 2 value is 0.96, and the results are shown in Fig. 16.
Model 3: optimizing the model for C 4 olens yield with catalyst combination and temperature The optimization model was established using eqn (6) and was constructed as follows: 35 The optimized conditions were divided according to the available experimental data: Lingo soware was used to identify the optimization solution, and the following results were obtained. Under charging method I, when T = 1.6, x1 = 0.5, and x2 = 1 (i.e. when the temperature was 450°C, the Co load was 0.5 wt%, and the Co/ SiO 2 was 200 mg), the maximum yield of C 4 olens was 52%.
When the temperature was below 350°C, the constraint conditions were changed to identify the optimal solution: 36 x2 $ 0.165; x2 # 1 Again using Lingo soware, the following results were obtained. When the temperature was lower than 350°C, T = 1.4, x1 = 0.5, and x2 = 1 (i.e. when the temperature was 350°C, the Co load was 0.5 wt%, and the Co/SiO 2 was 200 mg), the C 4 olens yield was at its maximum of 7.48%.
Using eqn (8) Lingo soware was again used to identify the optimization solution, and the following results were obtained. Under charging method II, when T = 1.6 and x1 = 4 (i.e. temperature was 400°C and Co/SiO 2 was 100 mg), the C4 olens yield reached the maximum of 23.67%. When the temperature was below 350°C and Co/SiO 2 was 100 mg, the C 4 olens yield reached the maximum of 9.92%.

Conclusion
In the preparation of C 4 olens through ethanol catalytic coupling, the ethanol conversion rate and C 4 olens selectivity are two core indexes. The results of this study indicate that the tting function between ethanol conversion rate, C 4 olens selectivity, and temperature under each catalyst combination predicted the values of ethanol conversion rate and C 4 olens selectivity under different temperatures. The two-factor analysis of variance showed that different catalyst combinations and temperatures had signicant effects on ethanol conversion rate and selectivity of C 4 olens. However, analysing the test results under a given catalyst combination at 350°C at different times in an experiment indicated that the ethylene selectivity and C 4 olens selectivity correlate less as reaction time increases. Therefore, the catalyst combination and reaction temperature are mainly considered when analysing the above indexes. To nd a certain catalyst combination and temperature that will achieve the highest C 4 olens yield under the same experimental conditions, a multiple nonlinear regression model and stepwise regression model of C 4 olens yield with four catalysts and temperatures were established and the goal function in the optimization model was obtained. Then, constraint conditions were given under laboratory conditions. Finally, the maximum C 4 olens yield was obtained.
Through the establishment and analysis of three mathematical models, this research showed that both catalyst combination and reaction temperature would affect the C 4 olens yield. Moreover, the higher the reaction temperature, the higher the yield of C 4 olens. The inuence of Co loading and Co/SiO 2 on the yield of C 4 olens is greater than that of the other two catalysts. When the minimum of Co loading was 0.5 wt% and the maximum of Co/SiO 2 was 200 mg, the yield of C 4 olens was largest. The amount of ethanol added per minute had little effect on the C 4 olens yield.
Based on the experimental data, this paper established a mathematical model and concluded that the higher the reaction temperature, the higher the C 4 olens yield. However, when the reaction temperature is higher than the maximum value of 400°C in the experimental data, will the C 4 olens yield continue to increase? And when the temperature continually rises, will the four catalysts undergo denaturation? There are insufficient experimental data to answer these questions, both of which need further study.

Data availability
The data in this paper come from Question B of the 2021 Chinese Contemporary Undergraduate Mathematical Contest in Modelling (CUMCM).

Author contributions
Conceptualization, methodology, soware, validation, formal analysis, data curation, writing-original dra preparation, visualization, P. T.; writing-review and editing, H. L.; supervision, project administration, funding acquisition, X. Z. and X. S. All authors have read and agreed to the published version of the manuscript.

Conflicts of interest
There are no conicts to declare.